I 



Length and Space. 69 



" equal to die same tiling, are equal to each other." This 

 i(eneral truth, we call an axiom or first truth ; because 

 it is certain, and cannot be called in question. Yet its 

 evidence arises entirely from the view of particular instan- 

 ces taken in detail ; and not from any quality in the 

 general proposition. This axiom is as applicable to num- 

 bers as to length, and its evidence there also arises from 

 a view of particular instances. By the application of 

 numbers to measures of length, we obtain another set of 

 axioms. Thus, things which are severaly double of one 

 thing, are equal. Things which are severally treble of one 

 tiling, are equal. Things which are halves of the same 

 thing, are equal. These axioms are obviously innumerable, 

 all undeniable, and of great utility. 



8. — Material objects may be considered as longer and 

 shorter, not only in regard to the distance of theone end from 

 the other, but also in regard to the distance of one side from 

 the other. For the sake of distinction, the former distance 

 is called the length, properly so called, of the object, and 

 the latter is termed its bkkautii. The two ideas are 

 obtained in perfectly the same way from touch, improved 

 in the same way by the sight, and perfected in the same 

 way by an accurate measure — they admit of the same 

 axioms. In fact, they differ in nothing except this single 

 circumstance, that the one is the distance of the ends, the 

 other of the sides. Whenever the length and breadtii are 

 unequal, the less is considered as the breadth, and the 

 greater as the length. Mathematicians correct the loose 

 ideas of mankind on this subject. But their (Ufiuition of 

 breadth pre-suppose.s that of a perpendicular, and previoUj, 

 even to that, the knowledge of the properties of u struight 

 line, which cannot be dcfuied. 



